Rust doesn't happen that fast so you will be fine
Be sure to keep driving it on wet roads AFTER they stop salting the roads to rinse off as much of the salt as possible
And/or put a garden sprinkler in the driveway and move the vehicle back and forth over it to do a final rinse
In below freezing temps rusting is very very very slow, its when it warms up that the residual salt can promote faster rusting
Rust is cause by Oxygen in the air, the process is called Oxidation
Bare metal(iron/steel) will oxidize when exposed to air, water(H2O) increases speed of oxidation, salt adds to that speed
Paint covers the metal so oxygen can't get to it, so no rust can start
Underside of any vehicle is constantly bombarded by road debris which wears paint/coatings off, exposing bare metal
Salt, or sand, on the road causes even more paint/coating wear
If you see rust spots then you need to neutralize the rust first, then you can paint/coat over it
"Naval Jelly" does that neutralizing, be careful using it, follow instructions
It changes Rust into a neutral material so paint will stick